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# Entropy

> [Lognormal][lognormal-distribution] distribution [differential entropy][entropy].

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<section class="intro">

The [differential entropy][entropy] (in [nats][nats]) for a [lognormal][lognormal-distribution] random variable is

<!-- <equation class="equation" label="eq:lognormal_entropy" align="center" raw="h\left( X \right) = \ln(\sigma e^{\mu +{\tfrac{1}{2}}}{\sqrt{2\pi}})" alt="Differential entropy for a lognormal distribution."> -->

<div class="equation" align="center" data-raw-text="h\left( X \right) = \ln(\sigma e^{\mu +{\tfrac{1}{2}}}{\sqrt{2\pi}})" data-equation="eq:lognormal_entropy">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/lognormal/entropy/docs/img/equation_lognormal_entropy.svg" alt="Differential entropy for a lognormal distribution.">
    <br>
</div>

<!-- </equation> -->

where `μ` is the location parameter and `σ > 0` is the scale parameter. According to the definition, the _natural logarithm_ of a random variable from a
[lognormal distribution][lognormal-distribution] follows a [normal distribution][normal-distribution].

</section>

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<section class="usage">

## Usage

```javascript
var entropy = require( '@stdlib/stats/base/dists/lognormal/entropy' );
```

#### entropy( mu, sigma )

Returns the [differential entropy][entropy] for a [lognormal][lognormal-distribution] distribution with location `mu` and scale `sigma` (in [nats][nats]).

```javascript
var y = entropy( 2.0, 1.0 );
// returns ~3.419

y = entropy( 0.0, 1.0 );
// returns ~1.419

y = entropy( -1.0, 2.0 );
// returns ~1.112
```

If provided `NaN` as any argument, the function returns `NaN`.

```javascript
var y = entropy( NaN, 1.0 );
// returns NaN

y = entropy( 0.0, NaN );
// returns NaN
```

If provided `sigma <= 0`, the function returns `NaN`.

```javascript
var y = entropy( 0.0, 0.0 );
// returns NaN

y = entropy( 0.0, -1.0 );
// returns NaN
```

</section>

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<section class="notes">

</section>

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<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var randu = require( '@stdlib/random/base/randu' );
var entropy = require( '@stdlib/stats/base/dists/lognormal/entropy' );

var sigma;
var mu;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    mu = ( randu()*10.0 ) - 5.0;
    sigma = randu() * 20.0;
    y = entropy( mu, sigma );
    console.log( 'µ: %d, σ: %d, h(X;µ,σ): %d', mu.toFixed( 4 ), sigma.toFixed( 4 ), y.toFixed( 4 ) );
}
```

</section>

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</section>

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<section class="related">

</section>

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<section class="links">

[lognormal-distribution]: https://en.wikipedia.org/wiki/Log-normal_distribution

[normal-distribution]: https://en.wikipedia.org/wiki/Normal_distribution

[entropy]: https://en.wikipedia.org/wiki/Entropy_%28information_theory%29

[nats]: https://en.wikipedia.org/wiki/Nat_%28unit%29

</section>

<!-- /.links -->
